
is called an interacting Brownian motion (in infinite dimensions) with potential
\( (\Phi , \Psi )\) . Here, \( (\nabla \Psi )(x,y)= \nabla _x \Psi (x,y)\) .
The study of interacting Brownian motions was initiated by Lang [1]}, [2]}, who
solved the above ISDE for \( (0, \Psi )\) , where \( \Psi \in C_0^3(\mathbb {R}^d)\)  is of Ruelle's class in the sense that it is super-stable and regular.
Fritz [3]} constructed non-equilibrium solutions for the same potentials as in [1]}, [2]} under the further restriction that the dimension \( d \le 4 \) .
Tanemura derived the solution for a hard-core potential [6]}, while
Fradon–Roelly–Tanemura solved the ISDE for the hard-core potential with long-range interactions, but still of Ruelle's class [7]}. Various ISDEs with logarithmic interaction potentials have also been solved [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}.
